On the Hardy-Littlewood-Sobolev type systems

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November 2016, 15(6): 2059-2074. doi: 10.3934/cpaa.2016027

On the Hardy-Littlewood-Sobolev type systems

Ze Cheng ^1, , Genggeng Huang ^2, and Congming Li ^3,

1.	Department of Applied Mathematics, University of Colorado at Boulder, Colorado
2.	Department of Mathematics, INS and MOE-LSC, Shanghai Jiao Tong University, Shanghai, China
3.	Department of Applied Mathematics, University of Colorado at Boulder

Received October 2015 Revised June 2016 Published September 2016

Abstract
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In this paper, we study some qualitative properties of Hardy-Littlewood-Sobolev type systems. The HLS type systems are categorized into three cases: critical, supercritical and subcritical. The critical case, the well known original HLS system, corresponds to the Euler-Lagrange equations of the fundamental HLS inequality. In each case, we give a brief survey on some important results and useful methods. Some simplifications and extensions based on somewhat more direct and intuitive ideas are presented. Also, a few new qualitative properties are obtained and several open problems are raised for future research.

Keywords: existence, uniqueness., non-existence, Hardy-Littlewood-Sobolev.

Mathematics Subject Classification: Primary: 35J9.

Citation: Ze Cheng, Genggeng Huang, Congming Li. On the Hardy-Littlewood-Sobolev type systems. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2059-2074. doi: 10.3934/cpaa.2016027

References:

[1]	F. Arthur, X. Yan and M. Zhao, A Liouville-type theorem for higher order elliptic systems,, \emph{Disc. & Cont. Dynamics Sys.}, 34 (2014), 3317. doi: 10.3934/dcds.2014.34.3317.
[2]	H. Berestycki, P. L. Lions and L. A. Peletier, An ODE approach to the existence of positive solutions for semi-linear problems in $R^N$,, \emph{Indiana University Mathematics Journal}, 30 (1981), 141. doi: 10.1512/iumj.1981.30.30012.
[3]	L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth,, \emph{Comm. Pure Appl. Math.}, 142 (1989), 615. doi: 10.1002/cpa.3160420304.
[4]	W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations,, \emph{Duke Math. J.}, 63 (1991), 615. doi: 10.1215/S0012-7094-91-06325-8.
[5]	W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, \emph{Commun. in Partial Differential Equations}, 30 (2005), 59. doi: 10.1081/PDE-200044445.
[6]	W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, \emph{Comm. Pure Appl. Math.}, 59 (2006), 330. doi: 10.1002/cpa.20116.
[7]	Z. Cheng, G. Huang and C. Li, A Liouville theorem for subcritical Lane-Emden system,, \arXiv{1412.7275}., ().
[8]	Z. Cheng and C. Li, Shooting method with sign-changing nonlinearity,, \emph{Nonlinear Analysis: Theory, 114 (2015), 2. doi: 10.1016/j.na.2014.10.019.
[9]	B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$,, \emph{Math. Anal. and Applications, 7A (1981), 369.
[10]	Y. Lei and C. Li, Sharp criteria of Liouville type for some nonlinear systems,, \emph{Discrete Contin. Dyn. Syst.}, 36 (2016), 3277. doi: 10.3934/dcds.2016.36.3277.
[11]	Y. Lei, C. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to the weighted HLS system,, \emph{Calc. Var. of Partial Differential Equations}, 45 (2012), 43. doi: 10.1007/s00526-011-0450-7.
[12]	C. Li and J. Villaver, Existence of positive solutions to semilinear elliptic systems with supercritical growth,, \emph{Comm. in Partial Differential Equation}, 41 (2016), 1029. doi: 10.1080/03605302.2016.1190376.
[13]	E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,, \emph{Ann. of Math.}, 118 (1983), 349. doi: 10.2307/2007032.
[14]	J. Liu, Y. Guo and Y. Zhang, Existence of positive entire solutions for polyharmonic equations and systems,, \emph{Journal of Partial Differential Equations}, 19 (2006).
[15]	E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $R^N$,, \emph{Quaderno Matematico}, (1982).
[16]	E. Mitidieri, A Rellich type identity and applications: Identity and applications,, \emph{Communications in partial differential equations}, 18 (1993), 125. doi: 10.1080/03605309308820923.
[17]	E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $R^N$,, \emph{Differ. Integral Equations}, 9 (1996), 465.
[18]	S. Pohozaev, Eigenfunctions of the equation $\Delta u + \lambda f(u)=0$,, \emph{Soviet Math. Doklady}, 6 (1965), 1408.
[19]	P. Polacik, P. Quittner and P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems, Part I: Elliptic equations and systems,, \emph{Duke Math. J.}, 139 (2007), 555. doi: 10.1215/S0012-7094-07-13935-8.
[20]	P. Pucci and J. Serrin, A general variational identity,, \emph{Indiana Univ. J. Math.}, 35 (1986), 681. doi: 10.1512/iumj.1986.35.35036.
[21]	Pavol Quittner and Philippe Souplet, Superlinear Parabolic Problems: Blow-Up, Global Existence and Steady States,, Springer, (2007).
[22]	J. Serrin, A symmetry problem in potential theory,, \emph{Arch. Rat. Mech. Anal.}, 43 (1971), 304.
[23]	J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems,, \emph{Differ. Integral Equations}, 9 (1996), 635.
[24]	J. Serrin and H. Zou, Existence of positive solutions of the Lane-Emden system,, \emph{Atti Semi. Mat. Fis. Univ. Modena}, 46 (1998), 369.
[25]	P. Souplet, The proof of the Lane-Emden conjecture in four space dimensions,, \emph{Advances in Mathematics}, 221 (2009), 1409. doi: 10.1016/j.aim.2009.02.014.
[26]	J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations,, \emph{Math. Ann.}, (1999), 207. doi: 10.1007/s002080050258.

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References:

[1]	F. Arthur, X. Yan and M. Zhao, A Liouville-type theorem for higher order elliptic systems,, \emph{Disc. & Cont. Dynamics Sys.}, 34 (2014), 3317. doi: 10.3934/dcds.2014.34.3317.
[2]	H. Berestycki, P. L. Lions and L. A. Peletier, An ODE approach to the existence of positive solutions for semi-linear problems in $R^N$,, \emph{Indiana University Mathematics Journal}, 30 (1981), 141. doi: 10.1512/iumj.1981.30.30012.
[3]	L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth,, \emph{Comm. Pure Appl. Math.}, 142 (1989), 615. doi: 10.1002/cpa.3160420304.
[4]	W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations,, \emph{Duke Math. J.}, 63 (1991), 615. doi: 10.1215/S0012-7094-91-06325-8.
[5]	W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, \emph{Commun. in Partial Differential Equations}, 30 (2005), 59. doi: 10.1081/PDE-200044445.
[6]	W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, \emph{Comm. Pure Appl. Math.}, 59 (2006), 330. doi: 10.1002/cpa.20116.
[7]	Z. Cheng, G. Huang and C. Li, A Liouville theorem for subcritical Lane-Emden system,, \arXiv{1412.7275}., ().
[8]	Z. Cheng and C. Li, Shooting method with sign-changing nonlinearity,, \emph{Nonlinear Analysis: Theory, 114 (2015), 2. doi: 10.1016/j.na.2014.10.019.
[9]	B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$,, \emph{Math. Anal. and Applications, 7A (1981), 369.
[10]	Y. Lei and C. Li, Sharp criteria of Liouville type for some nonlinear systems,, \emph{Discrete Contin. Dyn. Syst.}, 36 (2016), 3277. doi: 10.3934/dcds.2016.36.3277.
[11]	Y. Lei, C. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to the weighted HLS system,, \emph{Calc. Var. of Partial Differential Equations}, 45 (2012), 43. doi: 10.1007/s00526-011-0450-7.
[12]	C. Li and J. Villaver, Existence of positive solutions to semilinear elliptic systems with supercritical growth,, \emph{Comm. in Partial Differential Equation}, 41 (2016), 1029. doi: 10.1080/03605302.2016.1190376.
[13]	E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,, \emph{Ann. of Math.}, 118 (1983), 349. doi: 10.2307/2007032.
[14]	J. Liu, Y. Guo and Y. Zhang, Existence of positive entire solutions for polyharmonic equations and systems,, \emph{Journal of Partial Differential Equations}, 19 (2006).
[15]	E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $R^N$,, \emph{Quaderno Matematico}, (1982).
[16]	E. Mitidieri, A Rellich type identity and applications: Identity and applications,, \emph{Communications in partial differential equations}, 18 (1993), 125. doi: 10.1080/03605309308820923.
[17]	E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $R^N$,, \emph{Differ. Integral Equations}, 9 (1996), 465.
[18]	S. Pohozaev, Eigenfunctions of the equation $\Delta u + \lambda f(u)=0$,, \emph{Soviet Math. Doklady}, 6 (1965), 1408.
[19]	P. Polacik, P. Quittner and P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems, Part I: Elliptic equations and systems,, \emph{Duke Math. J.}, 139 (2007), 555. doi: 10.1215/S0012-7094-07-13935-8.
[20]	P. Pucci and J. Serrin, A general variational identity,, \emph{Indiana Univ. J. Math.}, 35 (1986), 681. doi: 10.1512/iumj.1986.35.35036.
[21]	Pavol Quittner and Philippe Souplet, Superlinear Parabolic Problems: Blow-Up, Global Existence and Steady States,, Springer, (2007).
[22]	J. Serrin, A symmetry problem in potential theory,, \emph{Arch. Rat. Mech. Anal.}, 43 (1971), 304.
[23]	J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems,, \emph{Differ. Integral Equations}, 9 (1996), 635.
[24]	J. Serrin and H. Zou, Existence of positive solutions of the Lane-Emden system,, \emph{Atti Semi. Mat. Fis. Univ. Modena}, 46 (1998), 369.
[25]	P. Souplet, The proof of the Lane-Emden conjecture in four space dimensions,, \emph{Advances in Mathematics}, 221 (2009), 1409. doi: 10.1016/j.aim.2009.02.014.
[26]	J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations,, \emph{Math. Ann.}, (1999), 207. doi: 10.1007/s002080050258.

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